Factor x^2+4x+5

Problem

x2+4*x+5

Solution

1. Identify the coefficients of the quadratic expression a*x2+b*x+c where a=1 b=4 and c=5

2. Calculate the discriminant using the formula D=b2−4*a*c to determine if the expression can be factored over the real numbers.

3. Substitute the values into the discriminant formula: D=4−4*(1)*(5)=16−20=−4

4. Conclude that since the discriminant is negative (D<0, the quadratic has no real roots and cannot be factored into linear factors with real coefficients.

5. Apply the quadratic formula x=(−b±√(,D))/(2*a) to find the complex roots: x=(−4±√(,−4))/(2*(1))=(−4±2*i)/2=−2±i

6. Write the factored form using the complex roots (r_1)=−2+i and (r_2)=−2−i in the form (x−(r_1))*(x−(r_2))

Final Answer

x2+4*x+5=(x+2−i)*(x+2+i)

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